International Journal of Theoretical Physics

, Volume 52, Issue 8, pp 2657–2673 | Cite as

Harmonic and Wave Maps Coupled with Einstein’s Gravitation

  • R. Schimming
  • Ragab M. Gad


In this paper we discuss the coupled dynamics, following from a suitable Lagrangian, of a harmonic or wave map ϕ and Einstein’s gravitation described by a metric g. The main results concern energy conditions for wave maps, harmonic maps from warped product manifolds, and wave maps from wave-like Lorentzian manifolds.


Harmonic map Wave map Energy conditions 



Former discussions and cooperations with T. Hirschmann and T. Deck are gratefully acknowledged. We thank the professors G. Huisken and A.D. Rendall for inspiring discussions.


  1. 1.
    Macias, A., Cervantes-Cota, J.L., Lämmerzahl, C.: Exact Solutions and Scalar Fields in Gravity. Kluwer, Dordrecht (2001) Google Scholar
  2. 2.
    Faraoni, V.: Cosmology in Scalar-Tensor Gravity. Kluwer, Dordrecht (2004) MATHCrossRefGoogle Scholar
  3. 3.
    Rendall, A.D.: Partial Differential Equations in General Relativity. Oxford University Press, Oxford (2008) MATHGoogle Scholar
  4. 4.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Eells, J., Lemaire, L.: Two Reports on Harmonic Maps. World Scientific, Singapore (1993) Google Scholar
  6. 6.
    Struwe, M.: Wave Maps. Birkhäuser, Basel (1997) Google Scholar
  7. 7.
    Bizon, P., Chmaj, T., Tabor, Z.: Dispersion and collapse of wave maps. Nonlinearity 13, 1411–1423 (2000) MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Bizon, P., Wasserman, A.: Self-similar spherically symmetric wave maps coupled to gravity. Phys. Rev. D 62, 084031 (2000) ADSCrossRefGoogle Scholar
  9. 9.
    Bizon, P., Wassermann, A.: On the existence of self-similar spherically symmetric wave maps coupled to gravity. Class. Quantum Gravity 19, 3309–3321 (2002) ADSMATHCrossRefGoogle Scholar
  10. 10.
    de Alfaro, V., Fubini, S., Furlan, G.: Gauge theories and strong gravity. Nuovo Cimento A 50, 523–554 (1979) ADSCrossRefGoogle Scholar
  11. 11.
    Omero, C., Percacci, R.: Generalized non-linear σ-models in curved space and spontaneous compactification. Nucl. Phys. B 165, 351–364 (1980) MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Gell-Mann, M., Zwiebach, B.: Spacetime compactification induced by scalars. Phys. Lett. B 141, 333–336 (1984) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Ghika, G.: Harmonic maps and submersions in local Euclidean gravity coupled to the σ-model. Rev. Roum. Phys. 31, 635–648 (1986) MathSciNetGoogle Scholar
  14. 14.
    Ghikaand, G., Corciovei, A.: Static solutions for the sigma-model coupled to gravity. Rev. Roum. Phys. 32, 827–835 (1987) ADSGoogle Scholar
  15. 15.
    Ghika, G., Visinescu, M.: Four-dimensional σ-model coupled to the metric tensor field. Nuovo Cimento B 59, 59–73 (1980) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Ianus, S., Visinescu, M.: Spontaneous compactification induced by non-linear dynamics. Class. Quantum Gravity 3, 889–896 (1986) MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Schimming, R., Hirschmann, T.: Harmonic maps from spacetimes and their coupling to gravitation. Astron. Nachr. 309, 311–321 (1988) MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Whitman, A.P., Knill, R.J., Stoeger, W.R.: Some harmonic maps on pseudo-Riemannian manifolds. Int. J. Theor. Phys. 25, 1139–1153 (1986) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    DeTurck, D.: Existence of metrics with prescribed Ricci curvature: local theory. Invent. Math. 65, 179–207 (1981) MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Friedrich, H., Rendall, A.D.: The Cauchy Problem for the Einstein Equations. Springer, Berlin (2000) Google Scholar
  21. 21.
    Lichnerowicz, A.: Radiations en relativite generale. In: Colloque de Royaumont, 1959. CNRS, Paris (1962) Google Scholar
  22. 22.
    Bel, L.: La radiation gravitationalle. In: Colloque de Royaumont, 1959. CNRS, Paris (1962) Google Scholar
  23. 23.
    Schimming, R.: Riemannsche Räume mit ebenfrontiger und mit ebener Symmetrie. Math. Nachr. 59, 129–162 (1974) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Fiedler, B., Schimming, R.: Exact solutions of the Bach field equations of general relativity. Rep. Math. Phys. 17, 15–36 (1980) MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Deck, T., Schimming, R.: Harmonic maps coupled to the Einstein equation. Universität Mannheim, (1991, preprint) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Universität GraeifswaldInstitut für Mathematik und informatikGreifswaldGermany
  2. 2.Mathematics Department, Faculty of ScienceKing Abdulaziz UniversityJeddahKSA
  3. 3.Mathematics Department, Faculty of ScienceMinia UniversityEl-MiniaEgypt

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